Performing 3D Similarity Transformation Using the Weighted Total Least-Squares Method

2014 ◽  
pp. 71-77
Author(s):  
J. Lu ◽  
Y. Chen ◽  
X. Fang ◽  
B. Zheng
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Similarity Transformation ◽  
Total Least Squares ◽  
Weighted Total Least Squares ◽  
3D Similarity

scholarly journals PERFORMING 3D SIMILARITY TRANSFORMATION WITH LARGE ROTATION ANGLES USING CONSTRAINED MULTIVARIATE TOTAL LEAST SQUARES

2020 ◽  
Vol 26 (4) ◽  
Author(s):  
Wuyong Tao ◽  
Xianghong Hua ◽  
Shaoquan Feng
Keyword(s):  
Least Squares ◽  
Similarity Transformation ◽  
Coefficient Matrix ◽  
Total Least Squares ◽  
Translation Vector ◽  
Large Rotation ◽  
Parameter Matrix ◽  
Transformation Parameters ◽  
Least Squares Algorithm ◽  
3D Similarity

Abstract: 3D similarity transformation is frequently encountered operation in the field of geodetic data processing, and there are many applications that involve large rotation angles. In previous studies, the errors of the coefficient matrix were usually neglected and a least squares algorithm was applied to calculate the transformation parameters. However, the coefficient matrix is composed of the point coordinates in source coordinate system, i.e., the coefficient matrix is also contaminated by errors. Therefore, a total least squares algorithm should be applied. In this paper, a new method is proposed to address the 3D similarity transformation problem with large rotation angles. Firstly, the scale factor and rotation matrix are put together as the parameter matrix to avoid the rank-defect problem. Then, the translation vector is removed and the multivariate model is constructed. Finally, the constraints are introduced according to the properties of the parameter matrix and the constrained multivariate total least squares algorithm is derived to obtain the transformation parameters. The experimental results show that the proposed method has a high computational efficiency.

scholarly journals On weighted total least squares adjustment for solving the nonlinear problems

2014 ◽  
Vol 4 (1) ◽  
Author(s):  
C. Hu ◽  
Y. Chen ◽  
Y. Peng
Keyword(s):  
Data Processing ◽  
Least Squares ◽  
Least Squares Method ◽  
Nonlinear Problems ◽  
Total Least Squares ◽  
Geodetic Data ◽  
Least Square ◽  
Weight Matrix ◽  
Weighted Total Least Squares ◽  
Least Squares Adjustment

AbstractIn the classical geodetic data processing, a non- linear problem always can be converted to a linear least squares adjustment. However, the errors in Jacob matrix are often not being considered when using the least square method to estimate the optimal parameters from a system of equations. Furthermore, the identity weight matrix may not suitable for each element in Jacob matrix. The weighted total least squares method has been frequently applied in geodetic data processing for the case that the observation vector and the coefficient matrix are perturbed by random errors, which are zero mean and statistically in- dependent with inequality variance. In this contribution, we suggested an approach that employ the weighted total least squares to solve the nonlinear problems and to mitigate the affection of noise in Jacob matrix. The weight matrix of the vector from Jacob matrix is derived by the law of nonlinear error propagation. Two numerical examples, one is the triangulation adjustment and another is a simulation experiment, are given at last to validate the feasibility of the developed method.

Fitting Weighted Total Least-Squares Planes and Parallel Planes to Support Tolerancing Standards

2012 ◽  
Author(s):  
Craig M. Shakarji ◽  
Vijay Srinivasan
Keyword(s):  
Least Squares ◽  
Point Contact ◽  
Coordinate Measuring Machine ◽  
Total Least Squares ◽  
Point Sampling ◽  
Least Squares Fitting ◽  
Weighted Total Least Squares ◽  
Measuring Machine ◽  
Fitting In ◽  
Value Decomposition

We present elegant algorithms for fitting a plane, two parallel planes (corresponding to a slot or a slab) or many parallel planes in a total (orthogonal) least-squares sense to coordinate data that is weighted. Each of these problems is reduced to a simple 3×3 matrix eigenvalue/eigenvector problem or an equivalent singular value decomposition problem, which can be solved using reliable and readily available commercial software. These methods were numerically verified by comparing them with brute-force minimization searches. We demonstrate the need for such weighted total least-squares fitting in coordinate metrology to support new and emerging tolerancing standards, for instance, ISO 14405-1:2010. The widespread practice of unweighted fitting works well enough when point sampling is controlled and can be made uniform (e.g., using a discrete point contact Coordinate Measuring Machine). However, we demonstrate that nonuniformly sampled points (arising from many new measurement technologies) coupled with unweighted least-squares fitting can lead to erroneous results. When needed, the algorithms presented also solve the unweighted cases simply by assigning the value one to each weight. We additionally prove convergence from the discrete to continuous cases of least-squares fitting as the point sampling becomes dense.

TOTAL LEAST SQUARES FOR ESTIMATION OF THE GROSS OUTPUT VECTOR

2020 ◽  
Author(s):  
Dmitriy Vladimirovich Ivanov ◽  
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Direct Costs ◽  
Total Least Squares ◽  
Test Cases ◽  
Output Vector ◽  
Final Consumption ◽  
The Matrix

The article proposes the estimation of the gross output vector in the presence of errors in the matrix of direct costs and the final consumption vector. The article suggests the use of the total least squares method for estimating the gross output vector. Test cases showed that the accuracy of the proposed estimates of the gross output vector is higher than the accuracy of the estimates obtained using the classical least squares method (OLS).

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